Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1221 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(16,24)(19,26)(22,28)(25,27);(6,9)(10,11)(13,14,17,20)(15,18,23,21)(16,19,24,26)(22,25,28,27);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28);(5,7,8,12)(6,10,9,11)(13,15)(14,18)(16,22)(17,23)(19,25)(20,21)(24,28)(26,27);(13,16)(14,19)(15,22)(17,24)(18,25)(20,26)(21,27)(23,28)Polycyclic presentation:
2^8|b^2=fh,c^2=g,d^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #113074, #113080, #113539, … |
| maximal normal subgroup of | #113074, #113080, #113539, … |
| maximal quotient of | #556363, #556365, #556375, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30544);